Breaking the 2 n barrier for 5-coloring and 6-coloring
BBreaking the 2 n barrier for 5-coloring and 6-coloring Or Zamir ∗ Abstract
The coloring problem (i.e., computing the chromatic number of a graph) can be solved in O ˚ p n q time, as shown by Bj¨orklund, Husfeldt and Koivisto in 2009. For k “ ,
4, better algorithms are knownfor the k -coloring problem. 3-coloring can be solved in O p . n q time (Beigel and Eppstein, 2005) and4-coloring can be solved in O p . n q time (Fomin, Gaspers and Saurabh, 2007). Surprisingly, for k ą O ˚ p n q are known. We show that both 5-coloring and 6-coloring can alsobe solved in O pp ´ ε q n q time for some ε ą
0. Moreover, we obtain an exponential improvement for k -coloring for any constant k for a very large family of graphs. In particular, for any constants k, ∆ , α ą k -coloring for graphs with at least α ¨ n vertices of degree at most ∆ can be solved in O pp ´ ε q n q time,for some ε “ ε k, ∆ ,α ą
0. As a consequence, for any constant k we can solve k -coloring exponentiallyfaster than O ˚ p n q for sparse graphs. The problem of k -coloring a graph, or determining the chromatic number of a graph (i.e., finding thesmallest k for which the graph is k -colorable) is one of the most classic and well studied NP-Completeproblems. Computing the chromatic number is listed as one of the first NP-Complete problems in Karp’spaper from 1972 [15]. In a similar fashion to k -SAT, the problem of 2-coloring is polynomial, yet k -coloringis NP-complete for every k ě The Exponential Time Hypothesis [13]), and thus it is believed that exact algorithms solving k -coloring must be exponential.There is a substantial and ever-growing body of work exploring exponential-time worst-case algorithms forNP-Complete problems. A 2003 survey of Woeginger [31] covers and refers to dozens of papers exploring suchalgorithms for many problems including satisfiability, graph coloring, knapsack, TSP, maximum independentsets and more. A subsequent book of Fomin and Kaski [10] further covers the topic of exact exponential-timealgorithms.For satisfiability (i.e., SAT), the running time of the trivial algorithm enumerating over all possible assign-ments is O ˚ p n q . No algorithms solving SAT in time O ˚ pp ´ ε q n q for any ε ą Strong
Exponential Time Hypothesis [5] states that no such algorithm exists. On theother hand, it is known that for every fixed k there exists a constant ε k ą k -SAT can be solvedin O ˚ pp ´ ε k q n q time. A result of this type was first published by Monien and Speckenmeyer in 1985 [21].A long list of improvements for the values of ε k were published since, including the celebrated 1998 PPSZalgorithm of Paturi, Pudl´ak, Saks and Zane [23] and the recent improvement over it by Hansen, Kaplan,Zamir and Zwick [11].For coloring, on the other hand, the situation is less understood. The trivial algorithm solving k -coloringby enumerating over all possible colorings takes O ˚ p k n q time. Thus, it is not even immediately clear thatcomputing the chromatic number of a graph can be done in O ˚ p c n q time for a constant c independent of k . ∗ Blavatnik School of Computer Science, Tel Aviv University, Israel. a r X i v : . [ c s . D S ] J u l n 1976, Lawler [19] introduced the idea of using dynamic-programming to find the minimal number ofindependent sets covering the graph. The trivial implementation of this idea results in an O ˚ p n q algorithm.More sophisticated bounds on the number of maximal independent sets in a graph and fast algorithms toenumerate over them (Moon and Moser [22], Paull and Unger [24]) resulted in an O ˚ p . n q algorithm.This was improved several times (including Eppstein [8] and Byskov [4]), until finally an algorithm computingthe chromatic number in O ˚ p n q time was devised by Bj¨orklund, Husfeldt and Koivisto in 2009 [3]. Thissettled an open problem of Woeginger [31]. A relatively recent survey of Husfeldt [12] covers the progresson graph coloring algorithms.For k “ ,
4, better algorithms are known for the k -coloring problem. Schiermeyer [27] showed that 3-coloringcan be solved in O ˚ p . n q time. Biegel and Eppstein [1] gave algorithms solving 3-coloring in O ˚ p . n q time and 4-coloring in O ˚ p . n q time in 2005. Fomin, Gaspers and Saurabh [9] have improved therunning time of 4-coloring to O ˚ p . n q in 2007. Unlike the situation in k -SAT, for every k ą k -coloring is O ˚ p n q , the same as computing the chromatic number. Thus, a veryfundamental question was left wide open. Open Problem 1.
Can -coloring be solved in O ˚ pp ´ ε q n q time, for some ε ą ? and more generally, Open Problem 2.
Can k -coloring be solved in O ˚ pp ´ ε k q n q time, for some ε k ą , for every k ? In our work, we answer Problem 1 affirmatively, the answer extends to 6-coloring as well. We also makemajor steps towards settling Problem 2.The main technical theorem of our paper follows.
Definition 1.1.
For ď α ď and ∆ ą we say that a graph G “ p V p G q , E p G qq is p α, ∆ q -bounded if itcontains at least α ¨ | V p G q| vertices of degree at most ∆ . Theorem 1.2.
For every k, ∆ , α ą there exists ε k, ∆ ,α ą such that we can solve k -coloring for p α, ∆ q -bounded graphs in O pp ´ ε k, ∆ ,α q n q time. In other words, we can answer Problem 2 affirmatively unless the graph has almost only vertices of super-constant degrees. In particular, we get faster algorithms for sparse graphs.
Corollary 1.3 (of Theorem 1.2) . For every k, C ą there exists ε k,C ą such that we can solve k -coloringfor graphs with | E p G q| ď C ¨ | V p G q| in O pp ´ ε k,C q n q time. Improvements for exponential algorithms solving partitioning problems in the case of bounded average degreeappeared before. Cygan and Pilipzcuk [7] obtained an improvement for the running time required for theTraveling Salesman Problem for graphs with bounded average degree. In their paper, they state the problemof doing the same for chromatic number as an open problem, which we settle in this paper. A similar resultfor the more restricted case of bounded degree graphs was obtained by Bj¨orklund et al. in [2].It is important to stress that Theorem 1.2 is much stronger than Corollary 1.3. In particular, we use it toconstruct the following reductions.
Theorem 1.4.
Given an algorithm solving p k ´ q -list-coloring in time O pp ´ ε q n q for some constant ε ą , we can construct an algorithm solving k -coloring in time O ` p ´ ε q n ˘ for some (other) constant ε ą . Furthermore, the reduction is deterministic. Theorem 1.5.
Given an algorithm solving p k ´ q -list-coloring in time O pp ´ ε q n q for some constant ε ą ,we can construct an algorithm solving k -coloring with high probability in time O ` p ´ ε q n ˘ for some (other)constant ε ą . From which we finally conclude the following, answering Problem 1 affirmatively.2 heorem 1.6. -coloring can be solved in time O pp ´ ε q n q for some constant ε ą . Theorem 1.7. -coloring can be solved with high probability in time O pp ´ ε q n q for some constant ε ą . We note that our 5-coloring algorithm is deterministic, while our 6-coloring algorithm is randomized withan exponentially small one-sided error probability.As part of our work, we develop a new removal lemma for small subsets. This could be of independentinterest. Very roughly, it states that every collection of small sets must have a large sub-collection that canbe made pairwise-disjoint by the removal of a small subset of the universe. The exact statement follows.
Theorem 1.8.
Let F be a collection of subsets of a universe U such that every set F P F is of size | F | ď ∆ .Let C ą be any constant. Then, there exist subsets F Ď F and U Ď U , such that • | F | ą ρ p ∆ , C q ¨ | F | ` C ¨ | U | , where ρ p ∆ , C q ą depends only on ∆ , C . • The sets in F are disjoint when restricted to U z U , i.e., for every F , F P F we have F X F Ď U . In Appendix A.1 we present an upper bound for the function ρ appearing in Theorem 1.8. This upper boundimplies that the constant ε we can obtain using our technique must be very small. The rest of the paper is organized as follows. In Section 2 we go over the preliminary tools that we use inthe paper. In Section 3 we further elaborate on the O ˚ p n q algorithm of Bj¨orklund, Husfeldt and Koivistofor computing the chromatic number of a graph [3].The main algorithmic contribution of the paper appears in Section 4, in which we prove Theorem 1.2. Thesection is partitioned into two main parts. In Section 4.1 we present our ideas in a simpler manner andget a result limited to bounded degree graphs. Then, in Section 4.2, which is more technically involved, wecomplete the proof of Theorem 1.2.As part of Section 4.2, we use Theorem 1.8, a combinatorial result independent of the algorithmic tools ofSection 4. The proof of Theorem 1.8 appears in Section 5.In Section 6 we use Theorem 1.2 as a main ingredient in a reduction from k -coloring to p k ´ q -list-coloring.In this section, we prove Theorems 1.4 and 1.6. In Section 7 we refine the ideas used in Section 6 andconstruct a reduction from k -coloring to p k ´ q -list-coloring. In this section, we prove Theorems 1.5 and 1.7.We finally conclude the paper and present a few open problems in Section 8. The terminology used throughout the paper is standard. For a graph G we denote by V p G q and E p G q its vertex-set and edge-set, respectively. For a subset V Ď V p G q we denote by G r V s the sub-graph of G induced by V . For v P V we denote by deg p v q the degree of v in G , by N p v q the set of neighbours of v , andby N r v s : “ N p V q Y t v u .For 0 ď α ď ą G “ p V p G q , E p G qq is p α, ∆ q -bounded if it contains at least α ¨ | V p G q| vertices of degree at most ∆. Note that if α “ k -coloring problem , we are given a graph G and need to decide whether there exists a k -coloring c : V p G q Ñ r k s of G , such that for every p u, v q P E p G q we have c p u q ‰ c p v q . If a graph has a k -coloring, wesay that it is k -colorable. In the chromatic number problem , we are given a graph G and need to compute χ p G q , the minimal integer k for which G is k -colorable.3n the k -list-coloring problem , we are given a graph G and a set C v Ď U of size | C v | ď k for every v P V p G q ,where U is some arbitrary universe. We need to decide whether there exists a coloring c : V p G q Ñ U suchthat for every v P V p G q we have c p v q P C v and for every p u, v q P E p G q we have c p u q ‰ c p v q .In a general p a, b q -CSP (Constraint Satisfaction Problem, see [18] or [28] for a complete definition anddiscussions) we are given a list of constraints on the values of subsets of size b of n a -ary variables, and needto decide whether there exists an assignment of values to the variables for which all constraints are satisfied. k -coloring and k -list-coloring are examples of p k, q -CSP problems. k -SAT is an example of a p , k q -CSPproblem. Let U be an n -element set. The Inverse M¨obius transform (sometimes also called the
Zeta transform) [26]maps a function f : P p U q Ñ R from the power-set of U into another function ˆ f : P p U q Ñ R defined asˆ f p X q “ ÿ Y Ď X f p Y q . Naively, ˆ f p X q is computed using 2 | X | additions. Thus, we can compute all values of ˆ f in a straightforwardmanner with O p n q operations. Yates’ method from 1937 ([17], [32]) improves on the above and computes allvalues of ˆ f using just O p n n q operations. The resulting algorithm is usually called the fast m¨obius transform or the fast zeta transform ([3], [16]). The authors of [2] and [3] use the fast Inverse M¨obius Transform todevise algorithms for combinatorial optimization problems such as computing the chromatic and the domaticnumbers of a graph. The algorithm of [3] is summarized in Section 3.A description of Yates’ method follows. Lemma 2.1.
The Inverse M¨obius Tranform ˆ f for some function f : P p U q Ñ R can be computed in O p n n q time, where n : “ | U | .Proof. Denote by U “ t u , . . . , u n u some enumeration of U ’s elements. Denote by f : “ f . We preform n iterations for i “ , . . . , n , in which we compute all values of the function f i : P p U q Ñ R defined using f i ´ as follows. f i p X q “ f i ´ p X q ` f i ´ p X zt u i uq if u i P Xf i ´ p X q otherwiseNamely, in the i -th iteration we add the values the function gets in the sub-cube defined by u i “ u i “ i shows that f i p X q “ ř Y P S i p X q f p Y q where S i p X q is the set of all subsets Y Ď X such that t u j P Y | j ą i u “ t u j P X | j ą i u In particular, by the end of the algorithm f n “ ˆ f . The k -coloring problem can be stated in two natural ways. In the first, given a graph G decide whether itcan be colored using k colors. The the second, given a graph G return a k -coloring for it if one exists, or saythat no such coloring exists. A few folklore reductions show that the two problems have the same runningtime up to polynomial factors. We state one for completeness. Others appear in the survey of [12]. A general constraint on a set x , . . . , x b of a -ary variables is a subset T of the a b possible assignments in t x , . . . x b u Ñ r a s .The constraint is satisfied by an assignment c , possibly on more variables, if c ˇˇ t x ,...x b u P T . emma 2.2. Let A be an algorithm deciding whether a graph is k -colorable in O p T p n qq time. Then, thereexists an algorithm A that finds a k -coloring for G , if one exists, in O ˚ p T p n qq time.Proof. We describe A . First, use A p G q to decide whether G is k -colorable, if it returns False we return thatno k -coloring exists. Otherwise, repeat the following iterative process. For every pair of distinct vertices p u, v q R E p G q that is not an edge of G , use A p G : “ p V p G q , E p G q Y tp u, v quqq to check whether G stays k -colorable after adding p u, v q as an edge. If it does, add p u, v q to E p G q . We stop when no such pair p u, v q exists.The reader can verify that the resulting graph must be a complement of k disjoint cliques, and thus we caneasily construct a k -coloring.A problem comes up while trying to use this type of reductions in the settings of this paper. The afore-mentioned reduction adds edges to the graph, and therefore increases the degrees of vertices. In particular,we cannot use it (or other similar reductions) in a black-box manner for statements like Theorem 1.2. Thealgorithm of [2] solves the decision version of k -coloring for bounded degree graphs, and cannot be triviallyconverted into an algorithm that finds a coloring. The algorithms presented in this paper, on the other hand,can be easily converted into algorithms that find a k -coloring. This is briefly discussed later in Section 4.3. O ˚ p n q algorithm In this section we present a summary of Bj¨orklund, Husfeldt and Koivisto’s algorithm from [3]. We presenta concise variant of their work that applies specifically to the coloring problem. The original paper covers alarger variety of set partitioning problems and thus the description in this section is simpler.We begin by making the following very simple observation, yielding an equivalent phrasing of the coloringproblem.
Observation 1.
A graph G is k -colorable if and only if its vertex set V p G q can be covered by k independentsets. A short outline of the algorithm follows, complete details appear below. We need to decide whether V p G q can be covered by k independent sets. In order to do so, we compute the number of independent sets in everyinduced sub-graph and then use a simple inclusion-exclusion argument in order to compute the number of(ordered) covers of V p G q by k independent sets. We are interested in whether this number is positive. Definition 3.1.
For a subset V Ď V p G q of vertices, let i p G r V sq denote the number of independent sets inthe induced sub-graph G r V s . We next show that using dynamic programming, we can quickly compute these values.
Lemma 3.2.
We can compute the values of i p G r V sq for all V Ď V in O ˚ p n q time.Proof. Let v P V be an arbitrary vertex contained in V . The number of independent sets in V that do notcontain v is exactly i p G r V zt v usq . On the other hand, the number of independent sets in V that do contain v is exactly i p G r V z N r v ssq . Thus, we have i p G r V sq “ i p G r V zt v usq ` i p G r V z N r v ssq . We note that both V zt v u and V z N r v s are of size strictly less than | V | . Thus, we can compute all 2 n valuesof i p G r¨sq using dynamic programming processing the sets in non-decreasing order of size.5onsider the expression F p G q “ ÿ V Ď V p G q p´ q | V p G q|´| V | ¨ i p G r V sq k . Using the values of i p G r¨sq computed in Lemma 3.2, we can easily compute the value of F p G q by directlyevaluating the above expression in O ˚ p n q time. Lemma 3.3.
Let S Ď S be sets. It holds that ÿ S Ď S Ď S p´ q | S | “ if S ‰ S p´ q | S | if S “ S Proof. If S Ĺ S then there exists a vertex v P S z S . We can pair each set S Ď S Ď S with S (cid:52) t v u , itssymmetric difference with t v u . Clearly, in each pair of sets one is of odd size and one is of even size, and thustheir signs cancel each other. Therefore, the sum is zero. In the second case, the claim is straightforward. Lemma 3.4. F p G q equals the number of k -tuples p I , . . . , I k ´ q of independent sets in G such that V p G q “ I Y . . . Y I k ´ .Proof. As i p G r V sq counts the number of independent sets in G r V s , raising it to the k -th power (namely, i p G r V sq k ) counts the number of k -tuples of independent sets in G r V s .Let p I , . . . , I k ´ q be a k -tuple of independent sets in G . It appears exactly in terms of the sum correspondingto sets V such that I Y . . . Y I k ´ Ď V Ď V p G q . Each time this k -tuple is counted, it is countedwith a sign determined by the parity of V . By Lemma 3.3, the sum of the signs corresponding to sets I Y . . . Y I k ´ Ď V Ď V p G q is zero if I Y . . . Y I k ´ ‰ V p G q and one if I Y . . . Y I k ´ “ V p G q .We conclude with Corollary 3.5. F p G q can be computed in time O ˚ p n q , and G is k -colorable if and only if F p G q ą . k -Coloring Algorithms for p α, ∆ q -bounded Graphs The main purpose of this section is proving Theorem 1.2.We first outline our approach. Let G be a graph with a constant chromatic number χ p G q ď k . It is wellknown that G must contain a large independent set. Let S be an independent set in G . We think of | S | asa constant fraction of | V p G q| , when we consider k as a constant. Let c : p V p G qz S q Ñ r k s be a k -coloringof the induced sub-graph G r V p G qz S s . We say that c can be extended to a k -coloring of G if there exists aproper k -coloring c : V p G q Ñ r k s such that c ˇˇ V p G qz S “ c . For a subset V Ď V p G qz S of vertices, we saythat c does not use the full palette on V if | c p V q| ă k , namely, if c does not use all k colors on the verticesof V . Clearly, a proper k -coloring c of V p G qz S can be extended to a proper k -coloring of G if and onlyif | c p N p s qq| ă k for every s P S .Our approach, on a high-level, is to construct an algorithm that finds anextendable k -coloring of V p G qz S . We aim to do so in O ´ | V p G qz S | p ´ ε q | S | ¯ time.6 𝑠 𝑠 𝑁(𝑠 )𝑁(𝑠 ) 𝑁(𝑠 ) In Section 4.1 we consider a restricted version of the problem in which the independent set S has the followingtwo additional properties. First, we assume that every vertex s P S is of degree deg p s q ď ∆, where ∆ issome constant. Second, we assume that no pair of vertices s , s P S share a neighbor in G . Equivalently,the neighborhoods N p s q for every s P S are all disjoint. Under these conditions, we present an algorithmthat runs in O ´ | V p G qz S | p ´ ε q | S | ¯ time, where ε depends only on ∆. As ε does not depend on k , we canin fact compute the chromatic number of G exponentially faster than O ˚ p n q if G contains an independentset S with these properties. We also observe that if G is of maximum degree ∆ then it contains a large suchindependent set S . Our algorithm is based on methods that generalize Section 3, and on a simple approachto implicitly compute values of the Inverse M¨obius Transform. 𝑠 𝑠 𝑠 𝑁(𝑠 )𝑁(𝑠 ) 𝑁(𝑠 ) In Section 4.2 we modify the algorithm of Section 4.1 and remove the second assumption on S . Namely, wenow only assume that S is an independent set and that for every s P S we have deg p s q ď ∆. Our algorithmstill runs in O ´ | V p G qz S | p ´ ε q | S | ¯ time, yet now ε depends on both ∆ and k . A main ingredient in themodification is a strong new removal lemma for small subsets. The proof of this combinatorial lemma isgiven in Section 5 and its statement is used in a black-box manner in this section. k -coloring bounded-degree graphs In this subsection we begin illustrating the ideas leading towards proving Theorem 1.2. We also prove thefollowing (much) weaker statement. 7 heorem 4.1.
For every k, ∆ there exists ε k, ∆ ą such that we can solve k -coloring for graphs withmaximum degree ∆ in O pp ´ ε k, ∆ q n q time. In fact, as a graph G with maximum degree ∆ has chromatic number χ p G q ď ∆ `
1, we can compute thechromatic number of a graph with degrees bounded by ∆ in time O pp ´ ε ∆ ` , ∆ q n q .As outlined in the beginning of this section, our approach begins by finding a large independent set with someadditional properties. We show that a graph with bounded degrees must contain a very large independentset S such that the distance between each pair of vertices in S is at least three. In other words, S is anindependent set, and no pair of vertices in S share a neighbor. In particular, the neighborhoods N p s q for s P S are all disjoint. The core theorem of this subsection is Theorem 4.2.
Let G be a graph and S Ď V p G q a set of vertices such that the distance between each twovertices in S is at least three and the degree of each vertex in S is at most ∆ . For any k , we can solve k -coloring for G in O ˚ ` | V p G q|´| S | ¨ p ´ ´ ∆ q | S | ˘ time. It is important to note that the existence of such a set S is our sole use of the bound on the graph degrees.Note that the bound of Theorem 4.2 does not depend on k . Thus, we get an exponential improvement forcomputing the chromatic number of a graph G that contains a large enough set S with the stated properties.Before proving Theorem 4.2, we describe a simple algorithm for finding a set S with the required propertiesin bounded-degree graphs. Lemma 4.3.
Let G be a graph with maximum degree at most ∆ . There exists a set S Ď V p G q of atleast ` ∆ ¨ | V p G q| vertices such that the distance between every distinct pair s , s P S is at least three.Furthermore, we can find such S efficiently.Proof. We construct S in a greedy manner. We begin with S “ H and V “ V p G q . As long as V is notempty we pick an arbitrary vertex v P V and add it to S . We then remove from V the vertex v and everyvertex of distance at most two from it.By construction, the minimum distance between a pair of vertices in S is at least three. The size of the2-neighborhood of a vertex is bounded by 1 ` ∆ ` ∆ ¨ p ∆ ´ q “ ` ∆ and thus we get the desired lowerbound on the size of S .Theorem 4.1 now follows from Lemma 4.3 and Theorem 4.2. Proof of Theorem 4.1.
Let G be a graph of maximum degree at most ∆ and let k be an integer. ByLemma 4.3, we can construct a set S of size | S | ě ` ∆ ¨ | V p G q| satisfying the conditions of Theorem 4.2.Thus, by Theorem 4.2, we can solve k -coloring for G in time O ˚ ´ n ´ ` ∆2 n ¨ p ´ ´ ∆ q ` ∆2 n ¯ “ O ˚ ˆˆ ¨ ´ ´ ´p ∆ ` q ¯ ` ∆2 ˙ n ˙ . In the rest of the subsection we prove Theorem 4.2.
Definition 4.4.
For subsets V Ď V p G qz S and S Ď S denote by β p V , S q the number of independent sets I in G r V s that intersect every neighborhood N p s q of s P S , that is, I X N p s q ‰ H for every s P S . Consider, for a subset S Ď S , the following sum h p G, S q : “ ÿ V Ď V p G qz S p´ q | V p G q|´| V | β ` V , S ˘ k . The following proof is almost identical to the proof of Lemma 3.4 in Section 3.8 emma 4.5. h p G, S q is the number of covers of V p G qz S by k -tuples p I , . . . , I k ´ q of independent sets in G r V p G qz S s such that I i X N p s q ‰ H for every s P S and every ď i ď k ´ .Proof. Each value of β p V , S q counts independent sets in G r V s that intersect every neighborhood N p s q for s P S .Each k -tuple p I , . . . , I k ´ q of that type is counted in terms corresponding to sets V such that I Y . . . Y I k ´ Ď V Ď V p G qz S. By Lemma 3.3 the multiplicity with which such k -tuple is counted is one if I Y . . . Y I k ´ “ V p G qz S. and zero otherwise.Consider the following expression. H p G, S q : “ ÿ S Ď S p´ q | S | h p G, S q H p G, S q is the number of covers of V p G qz S by k -tuples of independent sets that do not use the full palette on any neighborhood N p s q for s P S . The precise claim follows. Lemma 4.6. H p G, S q is the number of covers of V p G qz S by k -tuples p I , . . . , I k ´ q of independent sets in G r V p G qz S s such that for every s P S there exists ď i ď k ´ such that I i X N p s q “ H .Proof. In Lemma 4.5 we showed that h p G, S q counts the number of covers of V p G qz S by k -tuples p I , . . . , I k ´ q of independent sets in G r V p G qz S s such that for every s P S and for every 0 ď i ď k ´ I i X N p s q ‰ H .A covering k -tuple of independent sets p I , . . . , I k ´ q is counted exactly in terms corresponding to subsets S such that for every 0 ď i ď k ´ s P S , the independent set I i intersects the neighborhood N p s q . These are exactly the subsets S such that S Ď t s P S | @ ď i ď k ´ , I i X N p s q ‰ Hu . Using Lemma 3.3 with S “ H and S “ t s P S | @ ď i ď k ´ , I i X N p s q ‰ Hu we deduce that themultiplicity with which the k -tuple is counted is one if t s P S | @ ď i ď k ´ , I i X N p s q ‰ Hu “ H and zero otherwise.As outlined at the beginning of the section, we now claim that H p G, S q is positive if and only if G is k -colorable. Note that for the correctness of this lemma we still did not use the disjointness of the neighborhoods N p s q . We will need this property to improve the computation time. Lemma 4.7.
Let G be a graph and S an independent set in it. Then, H p G, S q ą if and only if G is k -colorable.Proof. Assume that there exists a k -coloring c : V p G q Ñ r k s of G . For 0 ď i ď k ´ I i : “ t v P V p G qz S | c p v q “ i u the subset of V p G qz S colored by i . Each I i is an independent set as c is a proper coloring of G . Furthermore,for each s P S , the neighborhood N p s q does not intersect I c p s q . Thus, p I , . . . , I k ´ q is a cover of V p G qz S by k independent sets that do not all intersect any neighborhood N p s q of s P S . By Lemma 4.6, H p G, S q ě H p G, S q ą k -coloring c : V p G qz S Ñ r k s of G r V p G qz S s such that the full palette is not used on anyneighborhood N p s q for s P S . Thus, we may extend c to a k -coloring c : V p G q Ñ r k s of the entire graph bycoloring each s P S with a color that does not appear in c p N p s qq . As S is an independent set, this coloringis proper.Up to this point, we have formalized the outline from the beginning of this section, reducing k -coloring to aproblem of k -coloring with some restrictions the smaller graph G r V p G qz S s and then to the computation of H p G, S q .Unfortunately, H p G, S q is a sum of 2 | S | terms, each of the form h p G, S q which is a sum of 2 | V p G q|´| S | termsby itself. Evidently, there are 2 n different terms of the form β p V , S q that are used in the definition of H p G, S q . Thus, we cannot hope to compute H p G, S q in less than 2 n steps if we need to explicitly examine2 n terms of the form β p¨ , ¨q . Moreover, it is also not clear how quickly we can compute the values of β p¨ , ¨q .We begin by explaining how values of β p¨q can be computed efficiently. The term h p G, S q is a weighted sumof the values β p V , S q for all V Ď V p G qz S . Denote by β µ p V , S q the indicator function that gets the value1 if V is an independent set in G r V p G qz S s and for every s P S we have V X N p s q ‰ H , and 0 otherwise.We can efficiently compute the value of β µ for a specific input in a straightforward manner (i.e., checkingwhether it is an independent set that intersects the relevant sets). We observe that β ` V , S ˘ “ ÿ V Ď V β µ ` V , S ˘ , thus, β “ ˆ β µ as functions of V , and we can compute the values of β p V , S q for all V Ď V p G qz S in O ˚ p | V p G q|´| S | q time using the Inverse M¨obius Transform presented in Section 2.1.An improvement to the running time comes from noticing that for many inputs p V , S q the value of β p V , S q is zero. In particular, if V X N p s q “ H , for some s P S , then β p V , S q “ V intersects N p s q . In the computation of h p G, S q we only need to consider termscorresponding to subsets V Ď V p G qz S in which for every s P S the intersection V X N p s q is non-empty, asthe values of other terms are all zero. We present a variant of the Inverse M¨obius Transform that computesonly the non-zero values by implicitly setting the others to zero. We then show that for most subsets S Ď S the number of non-zero entries is exponentially smaller than 2 | V p G q|´| S | . Definition 4.8.
For any S Ď S denote by B p S q : “ t V Ď V p G qz S | @ s P S . V X N p s q ‰ Hu the set ofall subsets of V p G qz S intersecting all neighborhoods of S . As we observed above, for every V R B p S q we have β p V , S q “
0. We conclude that
Observation 2.
For every S we have h p G, S q “ ÿ V P B p S q p´ q | V p G q|´| V | β ` V , S ˘ k . Lemma 4.9.
If the neighborhoods N p s q are disjoint for all s P S , then we can compute h p G, S q in O ˚ p| B p S q|q time.Proof. It suffices to compute β p V , S q for every V P B p S q and then use Observation 2. We do so byintroducing a variant of the Inverse M¨obius Transform that implicitly sets the value of β p V , S q to zero forevery V R B p S q .We first note that B p S q – P ˜ V p G qz ˜ S Y ď s P S N p s q ¸¸ ˆ ą s P S p P p N p s qqztHuq . r| B p S q|s and B p S q as a Cartesian product.We can also efficiently check if a set V belongs to B p S q . Let index : B p S q Ñ r| B p S q|s be a map from B p S q to indices of r| B p S q|s . If V R B p S q we define index p V q “ ´
1. By the observation above, we candefine index in way for which index and index ´ are efficiently computable. We also arbitrarily order thevertices of V p G qz S as v , v , . . . , v | V p G qz S | .We describe the algorithm in pseudo-code.Initialize an array f of size | B p S q| ; for (cid:96) in r| B p S q|s doif index ´ p (cid:96) q is an independent set in G r V p G qz S s then f p (cid:96) q Ð else f p (cid:96) q Ð for i in r| V p G qz S |s dofor (cid:96) in r| B p S q|s do V Ð index ´ p (cid:96) q ; if v i P V and index p V zt v i uq ‰ ´ then f p (cid:96) q Ð f p (cid:96) q ` f p index p V zt v i uqq ;We view f throughout the algorithm as function f : B p S q Ñ N . Denote the function represented by f atthe end of the first for loop by f . By definition, f p V q “ β µ p V , S q for every V P B p S q . Denote by f i the function represented by f at the end of the i -th iteration of the second (outer) for loop.We observe that f i is defined using f i ´ as f i p V q “ f i ´ p V q ` f i ´ p V zt v i uq if v i P V f i ´ p V q otherwisewhere f i ´ p V zt v i uq is implicitly defined to be zero if V zt v i u R B p S q .By induction on i , similar to this of Section 2.1, we can show that f i p V q “ ÿ V Ď V V zt v ,...,v i u“ V zt v ,...,v i u f p V q . In particular, by the end of the algorithm f “ ˆ f “ ˆ β µ “ β for the entire domain B p S q .After computing h p G, S q for every S Ď S we can compute H p G, S q in O ˚ p | S | q time. We thus finish theproof of Theorem 4.2 with the following counting lemma. Lemma 4.10.
Assume that the neighborhoods N p s q are disjoint for all s P S and that each neighborhood isof size | N p s q| ď ∆ . Then, ř S Ď S | B p S q| “ O ˚ ` | V p G qz S | ¨ p ´ ´ ∆ q | S | ˘ .Proof. Denote n p s q : “ | N p s q| . Also denote by N “ Ť s P S N p s q all neighbors of vertices of S and by N c “p V p G qz S q z N their complement in V p G qz S . We have | B p S q| “ | N c | ¨ ź s P S ´ n p s q ´ ¯ ¨ ź s P S z S n p s q “ | N c | ¨ ź s P S ´ ´ ´ n p s q ¯ ¨ ź s P S n p s q “ | N c | ¨ ź s P S ´ ´ ´ n p s q ¯ ¨ | N | “ | V p G qz S | ¨ ź s P S ´ ´ ´ n p s q ¯ . s P S we have n p s q ď ∆ and thus ` ´ ´ n p s q ˘ ď ` ´ ´ ∆ ˘ . Hence, | B p S q| ď | V p G qz S | ¨ ź s P S ` ´ ´ ∆ ˘ “ | V p G qz S | ¨ ` ´ ´ ∆ ˘ | S | . Therefore we have ÿ S Ď S | B p S q| ď ÿ S Ď S | V p G qz S | ¨ ` ´ ´ ∆ ˘ | S | “ | V p G qz S | ¨ | S | ÿ i “ ˆ | S | i ˙ ` ´ ´ ∆ ˘ i “ | V p G qz S | ¨ p ´ ´ ∆ q | S | . p α, ∆ q -bounded graphs In this section we prove the main technical theorem of the paper.
Theorem 1.2.
For every k, ∆ , α ą there exists ε k, ∆ ,α ą such that we can solve k -coloring for p α, ∆ q -bounded graphs in O pp ´ ε k, ∆ ,α q n q time. As in Section 4.1, we deduce Theorem 1.2 from the following theorem.
Theorem 4.11.
Let G be a graph and S Ď V p G q an independent set in G . Assume that the degree of eachvertex in S is at most ∆ . Then, we can solve k -coloring for G in O ˚ ` | V p G q| ¨ p ´ ε k, ∆ q | S | ˘ time, for someconstant ε k, ∆ ą . Let G be a graph with a subset U Ď V p G q of vertices such that for every v P U we have deg p v q ď ∆.In a similar fashion to Lemma 4.3 of the previous subsection (and even slightly simpler), we can greedilyconstruct a subset S Ď U of size | S | ě ` ∆ ¨| U | which is an independent set. Thus, Theorem 4.11 immediatelyimplies Theorem 1.2. Unlike the case of Section 4.1, this time the neighborhoods N p s q for s P S are notnecessarily disjoint. Thus, statements comparable to Lemma 4.10 are not true. Our solution for this problemis surprisingly general. In Section 1.8 we prove the following new type of removal lemma for small sets. Theorem 1.8.
Let F be a collection of subsets of a universe U such that every set F P F is of size | F | ď ∆ .Let C ą be any constant. Then, there exist subsets F Ď F and U Ď U , such that • | F | ą ρ p ∆ , C q ¨ | F | ` C ¨ | U | , where ρ p ∆ , C q ą depends only on ∆ , C . • The sets in F are disjoint when restricted to U z U , i.e., for every F , F P F we have F X F Ď U . Plugging F “ t N p s qu s P S , we get a small set U Ď V p G qz S of graph vertices, and a large subset S Ď S ofthe independent set, such that the neighborhoods N p s q of s P S become pairwise disjoint if we remove thevertices of U from G . As we want to preserve the correctness of the algorithm, we do not actually remove U from G , but enumerate over the colors they receive in a proper k -coloring, if one exists. The main technicalgap is adjusting the algorithm and proofs of Section 4.1 to the case in which some of the graph vertices havefixed colors. Theorem 4.12.
Let G be a graph, V Ď V p G q a subset of its vertices and c : V Ñ r k s a proper k -coloringof G r V s . Denote by V : “ V p G qz V . Let S Ď V be an independent set in G such that the distance in G r V s between each two vertices of S is at least three and the degree in G r V s of each vertex in S is atmost ∆ . For any k , we can decide whether c can be extended to a k -coloring of the entire graph G in O ˚ ` | V |´| S | ¨ p ´ ´ ∆ q | S | ˘ time. V p G q from V . Note that V doesnot include the vertices of V , as their colors are already fixed. For j P r k s , denote by V j : “ c ´ p j q thesubset of V colored by j color. Note that V “ Ť kj “ V j . We begin adapting the algorithm by redefiningthe β p¨ , ¨q function. Definition 4.13.
For subsets V Ď V z S , S Ď S , and a color j P r k s , we denote by β j p V , S q the numberof sets I Ď V such that I Y V j is an independent set in G and that I Y V j intersects N p s q for every s P S ,that is, for every s P S we have ´ I Y V j ¯ X N p s q ‰ H . We also revise the definition of h p G, S q : “ ÿ V Ď V z S p´ q | V |´| V | k ´ ź j “ β j ` V , S ˘ . The proof of Lemma 4.5 can be easily revised to show the following.
Lemma 4.14. h p G, S q is the number of covers of V z S by k -tuples of sets I , . . . , I k ´ such that for every j P r k s , I j Y V j is an independent set in G and that for every s P S and every j P r k s the set I j Y V j intersects the neighborhood N p s q . Without revising the definition of H p G, S q , the proof of Lemma 4.6 now shows that Lemma 4.15. H p G, S q is the number of covers of V z S by k -tuples of sets I , . . . , I k ´ such that for every j P r k s , I j Y V j is an independent set in G and that for every s P S the neighborhood N p s q is not intersectedby at least one of the k independent sets ´ I j Y V j ¯ for j P r k s . Therefore, we have
Lemma 4.16.
Let G be a graph, V Ď V p G q a subset of its vertices and c : V Ñ r k s a proper k -coloring of G r V s . Denote by V : “ V p G qz V . Let S Ď V be an independent set in G . Then, H p G, S q ą if and only if c can be extended to a k -coloring of G . The non-trivial part of the revision and the heart of this subsection, is adjusting the algorithm for computingthe values of h p G, S q without increasing the running time.For j P r k s , denote by S j : “ t s P S | N p s q X V j ‰ Hu the set of vertices in S whose neighborhood intersects V j . The key observation of this subsection follows. Lemma 4.17.
For any j P r k s , S Ď S , V Ď V , we have β j ` V , S ˘ “ β j ` V , S Y S j ˘ Proof.
For any set I Ď V the set I Y V j intersects every set in t N p s qu s P S j . In particular, an independentset I Ď V intersects all of t N p s qu s P S if and only if it intersects all of t N p s qu s Pp S Y S j q .Lemma 4.17 implies that it is enough to compute β j p V , S q only for sets S Ď S z S j , as its other values canbe deduced from these as β j p V , S q “ β j p V , S z S j q .For any S Ď S we again denote by B p S q : “ t V Ď V z S | @ s P S . V X N p s q ‰ Hu the set of all subsets of V z S intersecting all neighborhoods of S . Note the slight difference from Section 4.1 of considering subsetsof V z S and not of V p G qz S .As for every s P S z S j , N p s q X V j “ H , we still have that for every V R B p S q the value of β j p V , S q is zero.In particular, we can still use the implicit Inverse M¨obius Transform of Lemma 4.9 and get13 emma 4.18. Assume S Ď S z S j . We can compute β j p V , S q for every V P B p S q in O ˚ p| B p S q|q time. By Lemma 4.10 we get ÿ S Ď S z S j | B p S q| “ O ˚ ´ | V z S | ¨ p ´ ´ ∆ q | S z S j | ¯ . We can thus compute β j p V , S q for every S Ď S z S j and every V P B p S q in O ˚ ` | V z S | ¨ p ´ ´ ∆ q | S z S j | ˘ time. This is the time to emphasise a crucial point. Note that if we would consider every S Ď S insteadof S Ď S z S j , then the running time would be O ˚ ` | V z S | ¨ p ´ ´ ∆ q | S z S j | ¨ | S j | ˘ , as the neighborhoodscorresponding to S j are intersected by V j . This is why we compute every β j separately, and do so for all relevant sets S before computing even a single value h p G, S q . As it always holds that | S z S j | ď | S | , weconclude that Corollary 4.19.
We can compute β j p V , S q for all j P r k s , S Ď S z S j and V P B p S q in O ˚ ` | V z S | ¨ p ´ ´ ∆ q | S | ˘ time. Note that k “ O ˚ p q .We are now ready to compute the values of h p G, S q . We start by making the following observation. Observation 3. If Ş k ´ j “ S j ‰ H then c cannot be extended to a coloring of G . This holds as if some s P S has neighbors colored in each of the k colors then it cannot be properly colored.We are thus dealing with the case where Ş k ´ j “ S j “ H . Lemma 4.20.
For any S Ď S and V Ď V z S such that V R B p S q we have k ´ ź j “ β j ` V , S ˘ “ . Proof. As V R B p S q there exists some s P S such that V X N p s q “ H . As Ş k ´ j “ S j “ H , there exists a j P r k s for which s R S j . Thus, V j X N p s q “ H as well. We conclude that β j p V , S q “ h p G, S q : “ ÿ V P B p S q p´ q | V |´| V | k ´ ź j “ β j ` V , S z S j ˘ . Thus, we can compute h p G, S q in O ˚ p| B p S q|q time using the values computed in Corollary 4.19. UsingLemma 4.10 once again, we get that ÿ S Ď S | B p S q| “ O ˚ ´ | V z S | ¨ p ´ ´ ∆ q | S | ¯ which completes the proof of Theorem 4.12.We can now prove Theorem 4.11. Proof.
We apply the removal lemma of Theorem 1.8 to F “ t N p s qu s P S with C to be chosen later. We thusget a sub-collection S Ď S and a subset of vertices V Ď V p G qz S such that | S | ą ρ p ∆ , C q ¨ | S | ` C ¨ | V | andthat for every s , s P S it holds that N p s q X N p s q Ď V . Denote by V “ V p G qz p S Y V q . We enumerateover all k -colorings c : V Ñ r k s . For each coloring c , we check if it is a proper k -coloring of G r V s and if so14e apply Theorem 4.12 on G with V , c , S . If any of the applications of Theorem 4.12 returned that thereexists a valid extension of c to a coloring of G , we return that G is k -colorable, and otherwise that it is not.The running time of the entire algorithm, up to polynomial factors, is k | V | ¨ ´ | V z S | ¨ p ´ ´ ∆ q | S | ¯ “ | V | ¨ k | V | ¨ p ´ ´p ∆ ` q q | S | ď | V | ¨ k | V | ¨ p ´ ´p ∆ ` q q ρ p ∆ ,C q¨| S |` C ¨| V | . By picking C “ log k ´ log p ´ ´p ∆ ` q q ą k | V | ¨ p ´ ´p ∆ ` q q C ¨| V | “ | V | ¨ p ´ ´p ∆ ` q q ρ p ∆ ,C q¨| S | . In both previous subsections, we used the bounds on the degrees only in order to construct a good independentset S . After doing so, we may apply the self-reduction of Section 2.2 to the graph G r V p G qz S s , in whichwe no longer care about the number of edges nor the degrees. This would result in finding a k -coloring of G r V p G qz S s . Such coloring can be extended to a k -coloring of G by the constructive proof of Lemma 4.7.The exact claim follows. Lemma 4.21.
In the conditions of Theorem 4.2 or Theorem 4.11 we can also find a k -coloring of G .Proof. Consider the reduction between the decision and search versions of k -coloring of Lemma 2.2. Sinceadding edges to vertices whose both endpoints are in V p G qz S does not violate the conditions of the theorems,we may apply the reduction of Lemma 2.2 to G r V p G qz S s . By the end of the reduction, we have a k -coloringof G r V p G qz S s that is a restriction of some k -coloring of G . We can extend this k -coloring to a k -coloring of G using the algorithm of Lemma 4.7.As a corollary, in the conditions of Theorem 1.2 we can also find a k -coloring of G . In this section we show that any collection of small sets must contain a large sub-collection of almost pairwise-disjoint sets. The precise statement follows.
Theorem 1.8.
Let F be a collection of subsets of a universe U such that every set F P F is of size | F | ď ∆ .Let C ą be any constant. Then, there exist subsets F Ď F and U Ď U , such that • | F | ą ρ p ∆ , C q ¨ | F | ` C ¨ | U | , where ρ p ∆ , C q ą depends only on ∆ , C . • The sets in F are disjoint when restricted to U z U , i.e., for every F , F P F we have F X F Ď U . We should think of the statement of Theorem 1.8 in the following manner. We interpret almost pairwise-disjoint sub-collection as a sub-collection that would become pairwise-disjoint after the removal of a small number of elements of the universe. If ∆ is a constant, then the precise meaning of small and large is that onthe one hand, the size of the sub-collection is at least a constant fraction of the size of the entire collection,and on the other hand, its size is arbitrarily larger than the number of removed universe elements. Theconstant C represents the exact meaning of arbitrarily larger .15 efinition 5.1. For u P U , denote by deg p u q : “ |t F P F | u P F u| the number of sets in F containing it. We may think of our collection as a bipartite graph, where the left side consists of a vertex for each set in F ,the right side consists of a vertex for each element in the universe U , and every set F is connected to eachuniverse element it contains. Then, the degree of a universe element u is simply the degree of the vertexcorresponding to it in this graph.We begin by repeating and slightly generalizing Lemma 4.3, focusing on the case in which the universe hasbounded degrees. Lemma 5.2.
Assume that for every u P U we have deg p u q ď d , then we can construct a pair-wise disjoint sub-collection F Ď F of size F ě ¨ d | F | .Proof. We construct F in a greedy manner. Begin with F “ H being the empty set, and ˆ F : “ F beingthe entire collection. As long as ˆ F is non-empty, take an arbitrary element F out of it, add F to F , andremove every F P ˆ F intersecting F (including itself) from ˆ F . As each set F contains at most ∆ elementsand each element is contained in at most d sets, the number of sets intersecting a specific F is bounded by∆ ¨ d . Thus, the size of ˆ F can decrease by at most ∆ ¨ d after every step and therefore we manage to add atleast ¨ d | F | sets to F before ˆ F becomes empty.This leads to a very natural approach. As the number of high-degree universe elements should be small, wemay try removing all universe elements of degree above some threshold d and then use Lemma 5.2.As the sum of the degrees in each side of a bipartite graph is equal, we have ř u P U deg p u q “ ř F P F | F | ď ∆ ¨| F | .In particular, the number of universe elements of degree at least d is at most ∆ d | F | . Unfortunately, this islarger than ¨ d | F | , the size of the sub-collection we get by Lemma 5.2, even if we ignore C .On the other hand, we may notice that the worst-case collection for Lemma 5.2 is in fact not difficult to dealwith. Consider the case where the non-isolated vertices corresponding to universe elements are regular, i.e.,for each u P U we have deg p u q P t , d u for some d . In that case, the number of relevant universe elementsis indeed ∆ ¨| F | d , but after removing them, the entire collection becomes pair-wise disjoint. Thus, we mayeither get | F | “ ¨ d | F | and | U | “ | F | “ | F | and | U | “ ∆ ¨| F | d by removing the entirerelevant universe. It is easy to verify that for each C, d at least one of the two is large enough, in particular,for every d we get | F | ´ C ¨ | U | ě ` C ∆ | F | for at least one of them. Our proof captures this observation. Definition 5.3.
Denote by U p d q : “ |t u P U | deg p u q “ d u| the number of universe elements of degree d . The counting claim regarding the sum of the degrees in each side of the discussed bipartite graph can nowbe rephrased as
Corollary 5.4.
We have ř | F | d “ d ¨ U p d q ď ∆ | F | . Denote by V p d q : “ ¨ d | F | ´ C | F | ÿ i “ d ` U p i q Lemma 5.5.
For any d P N , we can construct a sub-collection F Ď F that is pair-wise disjoint after theremoval of a subset U Ď U of the universe, such that p| F | ´ C ¨ | U |q ě V p d q .Proof. We let U be the set of all universe elements of degree larger than d . Thus, | U | “ ř | F | i “ d ` U p i q . ByLemma 5.2, after the removal of U , we can construct a sub-collection of size | F | ě ¨ d | F | .By Lemma 5.5, in order to prove Theorem 1.8 it is enough to give a lower bound for max d V p d q that isproportional to | F | . Our approach for this maximization is analytical in nature, but we phrase it in adiscrete manner for simplicity.We remind the reader of the following well-known fact regarding the Harmonic series16 laim 5.6. Let H n : “ ř nk “ { k be the Harmonic series. Then, H n ě ln n ` γ , where γ ą . is theEuler-Mascheroni constant. Furthermore, for any n ě it holds that H t e n u ě n . We are now ready for the main Lemma needed for the proof of Theorem 1.8.
Lemma 5.7.
For any D P N , we have ř Dd “ V p d q ě ` ¨ | F | ¨ H D ´ C ∆ ¨ | F | ˘ .Proof. Consider the sum D ÿ d “ V p d q “ D ÿ d “ ¨˝ | F | ∆ ¨ d ´ C ¨ | F | ÿ i “ d ` U p i q ˛‚ “ | F | ∆ H D ´ C ¨ D ÿ d “ | F | ÿ i “ d ` U p i q We also notice that D ÿ d “ | F | ÿ i “ d ` U p i q ď | F | ÿ d “ | F | ÿ i “ d ` U p i q“ | F | ÿ d “ | F | ÿ i “ i ą d ¨ U p i q“ | F | ÿ i “ U p i q | F | ÿ d “ i ą d “ | F | ÿ d “ p i ´ q ¨ U p i qď | F | ÿ d “ i ¨ U p i q ď ∆ | F | where the last inequality follows from Corollary 5.4 and the indicator i ą d is defined to be 1 if i ą d and 0otherwise. Combining both inequalities we have D ÿ d “ V p d q “ | F | ∆ H D ´ C ¨ D ÿ d “ | F | ÿ i “ d ` U p i q ě | F | ∆ H D ´ C ∆ ¨ | F | proving the Lemma’s statement.By a standard averaging argument we have Observation 4.
For any D P N , max d V p d q ě D ř Dd “ V p d q . We are now ready to finish the proof of Theorem 1.8.
Proof.
By Observation 4 and Lemma 5.7, we have that for every D P N ,max d V p d q ě D ˆ | F | ∆ H D ´ C ∆ ¨ | F | ˙ . Slightly rearranging the inequality gives1 | F | max d V p d q ě D ˆ H D ´ C ∆ ˙ D “ t exp p ` C ∆ q u and using Claim 5.6 yields1 | F | max d V p d q ě exp p ` C ∆ q ˆ ` ` C ∆ ˘ ´ C ∆ ˙ ą ¨ e ` C ∆ . We complete the proof by using Lemma 5.5.We also observe that the sets F , U satisfying the statement of Theorem 1.8 can be computed efficiently. Observation 5.
Assuming ∆ , C are constants, sets F , U satisfying the statement of Theorem 1.8 can becomputed in time O p| F |q .Proof. Computing the bipartite graph representing the collection F takes linear time. Then, the proofs ofLemma 5.2 and Lemma 5.5 are both constructive and run in linear time for any specific degree threshold d .By the proof of Theorem 1.8 we notice that it is enough to consider only degree thresholds in the constant-sized range 1 ď d ď D “ t exp p ` C ∆ q u “ O p q . Thus, we may find a threshold satisfying the statement inlinear time.In Appendix A.1 we present a construction showing that in the settings of Theorem 1.8 we must have ρ p ∆ , C q ď p C ` q ´ ∆ . The construction is due to discussions with Noga Alon. k -coloring to p k ´ q -list-coloring In this section we use Theorem 1.2 in order to prove the following.
Theorem 1.4.
Given an algorithm solving p k ´ q -list-coloring in time O pp ´ ε q n q for some constant ε ą , we can construct an algorithm solving k -coloring in time O ` p ´ ε q n ˘ for some (other) constant ε ą . Furthermore, the reduction is deterministic. Beigel and Eppstein [1] show that 4-list-coloring (as a special case of a p , q -CSP) can be solved in time O p . n q . Therefore we conclude that Theorem 1.6. -coloring can be solved in time O pp ´ ε q n q for some constant ε ą . We begin by illustrating the idea intuitively. By Theorem 1.2, it suffices to solve k -coloring for graphs inwhich most vertices have high degrees. We show that in this case, the graph has a small dominating set ,this is a subset R of vertices such that every vertex not in R is adjacent to at least one vertex of R . Givena k -coloring of the dominating set, the problem of extending the coloring to a k -coloring of the entire graphbecomes a problem of p k ´ q -list-coloring the rest of the graph. This is because each vertex not in thedominating set has a neighbor in it, and thus has at least one of the k colors which it cannot use. Assumingthe dominating set is small enough, we can enumerate over the k -colorings of vertices in it, and then solvethe remaining p k ´ q -list-coloring problem. Lemma 6.1.
Let G be a graph. Assume that there exists a subset of vertices V Ď V p G q of size | V | ěp ´ α q ¨ | V p G q| such that for every v P V we have deg p v q ě ∆ ´ . Then, G has a dominating set R Ď V p G q of size | R | ď ` p ´ α q ¨ ` ln ∆∆ ` α ˘ ¨ | V p G q| . Furthermore, there is an efficient deterministic algorithm tofind such a dominating set.Proof. Denote by δ p G q the minimum degree of a vertex in G . Let R be a random subset of V p G q chosenby picking each v P V p G q independently with probability p . We have E r| R |s “ p ¨ | V p G q| . Let v P V be avertex of degree at least ∆ ´
1, the probability of not adding v or any one of its neighbors to R is at most p ´ p q ∆ . Denote by R the set of vertices that are in V but not in R and do not have a neighbor in R .18y the previous observation, E r| R |s ď p ´ p q ∆ ¨ | V | . Similarly, denote by R the set of vertices that arein V p G qz V , not in R and do not have a neighbor in R . We have E r| R |s ď p ´ p q δ p G q` ¨ | V p G qz V | .The set R “ R Y R Y R is a dominating set. We have E r| R |s ď E r| R |s ` E r| R |s ` E r| R |sď p ¨ | V p G q| ` p ´ p q ∆ ¨ | V | ` p ´ p q δ p G q` ¨ | V p G qz V |ď ´ p ` p ´ α q ¨ p ´ p q ∆ ` α ¨ p ´ p q δ p G q` ¯ ¨ | V p G q| . Furthermore, we can efficiently compute E r| R |s even after conditioning on whether or not vertices are chosento R . Thus, the method of conditional expectations results in an efficient deterministic algorithm that findsa set R of size at most the above expectation. We elaborate on the matter in Appendix A.2.While not optimal for many parameters, for the sake of our use of this lemma it suffices to pick p “ ln ∆∆ , forwhich we get | R | ď ˆ ln ∆∆ ` p ´ α q ¨ p ´ ln ∆∆ q ∆ ` α ¨ p ´ ln ∆∆ q δ p G q` ˙ ¨ | V p G q|ď ˆ ln ∆∆ ` p ´ α q ¨ e ´ ln ∆ ` α ¨ p ´ ln ∆∆ q ˙ ¨ | V p G q|“ ˆ p ´ α q ¨ ` ln ∆∆ ` α ˙ ¨ | V p G q| . As evident in the proof of Lemma 6.1, a lower bound on the minimum degree δ p G q of the graph can resultin a slightly better bound on the size of the dominating set we can construct. While it is not necessary forproving the statement of Theorem 1.4, we include the following observation for completeness. Lemma 6.2.
Given an algorithm solving k -coloring for graphs of minimum degree δ p G q ě k , we can con-struct an algorithm solving k -coloring for every graph with the same running time (up to an additive poly-nomial factor).Proof. Denote by A the algorithm solving k -coloring for graphs with minimum degree δ p G q ě k . Given agraph G , we initiate a stack σ and run the following iterative process. As long as there is a vertex v in G of degree deg p v q ă k , we push v into σ and remove it and its adjacent edges from G . When we finish, ourgraph is of minimal degree δ p G q ě k and thus we can run A . If G , which is currently an induced sub-graphof the input graph, is not k -colorable, then the input graph is not k -colorable as well. Otherwise, we extendthe coloring c of G returned by A iteratively as follows. As long as σ is not empty, pop a vertex v out of it.Re-insert v and its adjacent edges back into G . As by construction it is of degree deg p v q ă k , we must haveat least one color i that is not used for any of v ’s neighbors. Extend c to v by setting c p v q “ i . When thestack σ is empty, c is a k -coloring of the entire input graph. Lemma 6.3.
Let G be a graph with a dominating set R . We can solve k -coloring for G by solving k | R | instances of p k ´ q -list-coloring on graphs with | V p G q| ´ | R | vertices.Proof. A k -coloring c : R Ñ r k s of G r R s can be extended to a k -coloring c : V p G q Ñ r k s of G with c | R “ c ,if and only if there is valid coloring of G r V p G qz R s such that a vertex v P V p G qz R can only be colored witha color from r k sz c p R X N p v qq . As each v P V p G qz R has at least one neighbor in R , we have | R X N p v q| ě |r k sz c p R X N p v qq| ď k ´
1. Thus, we are left with a p k ´ q -list-coloring problem on G r V p G qz R s . 19e are now ready to prove Theorem 1.4. Proof.
Let ∆ , α ą G with n vertices, we check whether itis p α, ∆ q -bounded. If it is, then we use the algorithm of Theorem 1.2 to solve k -coloring in O pp ´ ε α, ∆ q n q time. Otherwise, there are more than p ´ α q n vertices of degree larger than ∆ and by Lemma 6.1 wecan find a dominating set R of G of size | R | ď ` p ´ α q ¨ ` ln ∆∆ ` α ˘ n . Using Lemma 6.3 and the given p k ´ q -list-coloring algorithm, we can solve k -coloring for G in time k | R | ¨ p ´ ε q n ´| R | “ ˆ k ´ ε ˙ | R | ¨ p ´ ε q n ď ˆ k ´ ε ˙ p p ´ α q¨ ` ln ∆∆ ` α q n ¨ p ´ ε q n . Combining both cases, we get an algorithm running in time O p ´ ε q n for ε : “ min ˜ ε α, ∆ , ´ ˆ k ´ ε ˙ p p ´ α q¨ ` ln ∆∆ ` α q ¨ p ´ ε q ¸ . When α Ñ Ñ 8 the second expression converges to ε ą
0. Therefore, for any choice of a smallenough constant α and large enough integer ∆ we have ε ą k -coloring to p k ´ q -list-coloring We now refine the reduction of Section 6 and show that
Theorem 1.5.
Given an algorithm solving p k ´ q -list-coloring in time O pp ´ ε q n q for some constant ε ą ,we can construct an algorithm solving k -coloring with high probability in time O ` p ´ ε q n ˘ for some (other)constant ε ą . Once again, we use the 4-list-coloring algorithm of Beigel and Eppstein [1] to conclude
Theorem 1.7. -coloring can be solved with high probability in time O pp ´ ε q n q for some constant ε ą . We begin by outlining the way in which the previous reduction can be improved. Consider the reduction ofSection 6 and specifically the proof of Theorem 1.4. In the case where the graph is p α, ∆ q -bounded, we maystill use Theorem 1.2 and gain an exponential improvement. We now focus on the other case, in which mostvertices are of degrees larger than ∆. Let ∆ be some constant to be chosen later. We think of ∆ as largeyet arbitrarily smaller than ∆. Consider an arbitrary k -coloring c : V p G q Ñ r k s of G . For a vertex v P V p G q we denote by N i p v q : “ N p v q X c ´ p i q the set of v ’s neighbors that are colored by i in c . We say that a vertex v is good if there are at least two distinct colors i ‰ j for which | N i p v q| , | N j p v q| ą ∆ . As in the proof ofLemma 6.1, a small random subset of vertices (whose size depends on ∆ ) is likely to hit at least one neighborof v of color i and at least one neighbor of v of color j . Denote by β the fraction of bad (i.e., not good )vertices in V p G q . If β is small enough, a reduction almost identical to the previous one works. Uniformallypick a random small set R of graph vertices, enumerate over the colorings of the vertices in R . In one ofthe colorings (the one corresponding to c restricted to R ) we expect having in R neighbors of at least twodifferent colors for almost all vertices of V p G qz R . With a cautious implementation, this gives a reductionto p k ´ q -list-coloring. Thus, the interesting case is when α is very small yet β is large. For a bad vertex v P V p G q of degree larger than ∆, we must have single color i such that | N i p v q| ě ´ ´ p k ´ q¨ ∆ ∆ ¯ | N p v q| .Thus, almost all of the neighbors of a bad vertex can be colored by the same color. We therefore aim to gainby picking a large subset of v ’s neighbors and contract them to a single vertex. It is likely that c remains avalid coloring after the contraction. Furthermore, if the contracted set is an independent set, a coloring of theresulting graph is also a coloring of the original graph. The algorithmic harnessing of the above observationis somewhat involved, as we cannot identify good and bad vertices easily.20ick R , a random subset of V where vertices are picked i.i.d. with probability ln ∆ ∆ ; if | R | ą ` ln ∆ ∆ n then Return that no coloring was found and halt; for
Every function c : R Ñ r k s doif c is a valid coloring of G r R s then Compute R : “ B p R , c q ; if | R | ă βn ` n thenfor Every function c : R Ñ r k s doif c Y c is a valid coloring of G r R Y R s thenfor v P V p G qzp R Y R q do L p v q : “ r k szp c Y c q p N p v q X p R Y R qq ;Run A on V p G qzp R Y R q with the lists L p¨q ;If it returns a coloring c , return c Y c Y c and halt;Return that no coloring was found; Algorithm 1:
Algorithm A p G, k, ∆ , β q Lemma 7.1.
Let G be a k -colorable graph, ∆ be some constant, let c be a k -coloring of G . Assume thatwe are also given β , an upper bound on the fraction of bad vertices in G with respect to c, ∆ . Given analgorithm A solving p k ´ q -list-coloring in time O pp ´ ε q n q , we can construct an algorithm A that runs in O ¨˝ˆ k ´ ε ˙ ´ ` ln ∆ ∆ ` β ¯ n ¨ p ´ ε q n ˛‚ time, and returns a k -coloring of G with probability at least .Proof. Let R be a random subset of G ’s vertices, picking each vertex independently with probability p . Let v be a good vertex and i, j two colors for which | N i p v q| , | N j p v q| ą ∆ . We have a probability of at most2 ¨ p ´ p q ∆ that either N i p V q X R or N j p v q X R is empty. Thus, the expected number of good verticeswithout neighbors in R of two different colors (according to c ) is bounded by 2 p ´ p q ∆ n . By Markov’sinequality, with probability greater than their number is at most 5 p ´ p q ∆ n . For any R Ď V p G q anda partial coloring c : R Ñ r k s , denote by B p R , c q the set of all vertices in V p G qz R that do not haveneighbors of two different colors (according to c ) in R . By the above, we have that | B p R , c ˇˇ R q| ď βn ` p ´ p q ∆ n with probability at least . We pick p “ ln ∆ ∆ and have βn ` ¨ p ´ p q ∆ ¨ n ă βn ` n. We also note that | R | „ Bin p n, p q “ Bin p n, ln ∆ ∆ q . Thus by applying the standard Chernoff boundPr p X ą p ` δ q µ q ă e ´ δ µ with δ “ we have P r p| R | ą ` ln ∆ ∆ n q ă e ´ n ln ∆ ă . We therefore consider Algorithm 1.The correctness is quite straightforward. Every coloring returned by the algorithm is valid, and with prob-ability at least we reach the inner for loop with both c Y c “ c ˇˇ R Y R and thus A will return a validsolution. The inner loops run at most k ` ln ∆ ∆ n ` βn ` n times and thus we get the desired running time.21 f G is p α , ∆ q -bounded then Run Algorithm A k, ∆ ,α p G q ; else Run Algorithm A p G, k, ∆ , β q ; Algorithm 2:
Algorithm A p G, k, ∆ , ∆ , α , β q If we choose a large enough constant ∆ and β is small enough, Lemma 7.1 gives an exponential improvement.We next deal with the case where β is not small enough, and then finally discuss our concrete algorithm(that cannot compute or use the value of β ). Lemma 7.2.
Let G be a k -colorable graph, ∆ , r ě be some integers, and let c be a k -coloring of G . Let β be a lower bound on the fraction of bad vertices in G with respect to c, ∆ . Denote by ∆ : “ r p k ´ q ∆ ` r and by α the fraction of G ’s vertices of degrees at most ∆ .If we pick a random vertex v P V p G q and then a random subset S Ď N p v q of size exactly r , then theprobability that c p u q is identical for all u P S is at least p β ´ α q .Proof. With probability at least β ´ α the vertex v is bad and of degree larger than ∆. In this case, thereexists a single color i such that for all j ‰ i we have | N j p v q| ă ∆ . We construct S iteratively by picking arandom neighbor of v that is not already in S for r times. After (cid:96) ă r iterations, the probability of a randomvertex of N p v qz S to be in N i p v q is at least | N i p v q| ´ r | N p v q| “ ´ r ` ř j ‰ i | N j p v q|| N p v q| ě ´ r ` p k ´ q ¨ ∆ ∆ “ ´ r . Thus, the probability that all r neighbors are in N i p v q is at least ˆ ´ r ˙ r ě . Intuitively, if p β ´ α q ą ´p r ´ q it is beneficial to use Lemma 7.2 and contract the set S , decreasing thenumber of vertices by p r ´ q .For constants ∆ , r we set ∆ : “ r ` r p k ´ q ∆ β : “ ¨ p ´ ε q ´p r ´ q ´ p ´ ε q ´ r α : “ β . Furthermore, we pick ∆ , r to be large enough to satisfy ˆ k ´ ε ˙ ´ ` ln ∆ ∆ ` β ¯ ¨ p ´ ε q ă . Let ε be ˜ ´ ´ k ´ ε ¯ ´ ` ln ∆ ∆ ` β ¯ ¨ p ´ ε q ¸ P p , ε q . Let ε the minimum between ε k, ∆ ,α of Theorem 1.2and ε .We first combine Algorithm A k, ∆ ,α of Theorem 1.2 and Algorithm A of Lemma 7.1 and define Algorithm 2that covers both the case when β is small and the case when α is large.The following Lemma immediately follows 22et flag to 1 with probability p ´ ε q ´| V p G q| , and to 0 otherwise; if flag is or | V p G q| ď r then Run Algorithm A p G, k, ∆ , ∆ , α , β q and return its output; else Choose a random v uniformly out of V p G q ; if deg p v q ă ∆ then Halt;Choose uniformally a random subset S Ď N p v q of size exactly r ; if S is not an independent set in G then Halt;Contract S to a single vertex in G ;Run A p G q recursively. If the recursive call returned a coloring, we convert it to a coloring of theoriginal graph by expanding the contracted vertex back into S and giving all of its vertices thecolor of the contracted vertex; Algorithm 3:
Algorithm A p G, k q Lemma 7.3.
Algorithm A runs in O ` p ´ ε q n ˘ time, and if p β ´ α q ď p β ´ α q it returns a k -coloring of G with probability at least .Proof. If α ě α then we run Algorithm A k, ∆ ,α and thus correctness follows from Theorem 1.2. Otherwise, β ď α ` p β ´ α q ă β and thus correctness follows from Lemma 7.1.We finally prove Theorem 1.5 by constructing Algorithm 3. Lemma 7.4. If G is k -colorable then A p G q returns a coloring with probabilityat least p ´ ε q ´p| V p G q|` q .Proof. We prove the claim by induction on | V p G q| . The base case | V p G q| ď r follows from the correctnessof Algorithm A . We now prove the induction step by considering two cases. If p β ´ α q ď p β ´ α q thenwith probability p ´ ε q ´| V p G q| we set flag to 1 and run Algorithm A . We then produce a coloring withprobability at least by Lemma 7.3. Otherwise, with probability 1 ´ p ´ ε q ´| V p G q| ě ´ p ´ ε q ´ r we set flag to 0. We have 14 p β ´ α q ą p β ´ α q “ β “ p ´ ε q ´p r ´ q ´ p ´ ε q ´ r , and thus by Lemma 7.2 we both set flag to 0 and pick a set S such that G remains k -colorable after thecontraction with probability greater than p ´ ε q ´p r ´ q . The contraction decreases | V p G q| by exactly r ´ p ´ ε q ´p| V p G q|´p r ´ q` q . In both cases, the induction hypothesis holds for | V p G q| . Lemma 7.5.
The expected running time of Algorithm A is O ´´ ´ ε ´ ε ¯ n ¯ .Proof. With probability p ´ ε q ´| V p G q| we set flag to 1 and run Algorithm A which takes O ´ p ´ ε q | V p G q| ¯ time. Otherwise, we recursively run A on a graph with | V p G q|´p r ´ q vertices. Thus, the expected running23ime is T p n q “ p ´ ε q ´ n ¨ O `` ´ ε ˘ n ˘ ` ´ ´ p ´ ε q ´ n ¯ ¨ T p n ´ p r ´ qqď O ˆˆ ´ ε ´ ε ˙ n ˙ ` T p n ´ p r ´ qq“ . . . “ O ˆˆ ´ ε ´ ε ˙ n ˙ . The last equality holds as ε ă ε . Proof of Theorem 1.5.
We run A p G q for n ¨ p ´ ε q n ` times. If any of them found a coloring we returnit and otherwise say that the graph is not k -colorable. If G is k -colorable, the probability we never find acoloring is bounded by ´ ´ p ´ ε q ´p n ` q ¯ n ¨p ´ ε q n ` ă e ´ n . The expected running time of all iterations together is O ` n p ´ ε q n ˘ . We can terminate the run of thealgorithm if it takes much longer than its expected run-time as with high probability it does not happen. The main algorithmic contribution of the paper is Theorem 1.2. We use it in order to answer a few fun-damental questions regarding the running time of k -coloring algorithms. In particular, we present the first O pp ´ ε q n q algorithms solving 5-coloring and 6-coloring, for some ε ą
0. While the ε we can get using ourtools is very small, this serves as the first proof that 5-coloring can be solved faster than we can currentlycompute the chromatic number in general. The upper bound in Appendix A.1 shows that the magnitude of ε is a necessary consequence of using the removal lemma.The main open problem that we leave unsettled is Open Problem 3.
Can we solve k -coloring in O ˚ pp ´ ε k q n q time for some ε k ą , for every k ? Theorem 1.2 makes some progress towards answering it, by giving some additional conditions on the inputgraph under which the answer is affirmative. In particular, we show that it holds for sparse graphs, i.e.,graphs of bounded average degree. A new problem arising from this observation is
Open Problem 4.
Can we compute the chromatic number in O ˚ pp ´ ε d q n q time for graphs with averagedegree bounded by d , for some ε d ą and every d ? We note that in Theorem 4.2, which is a somewhat simplified version of Theorem 1.2, we do not require k to be a constant (i.e., we get an exponential improvement for computing the chromatic number of suchgraphs). It is intriguing to understand if the assumption that k is constant is inherent in Theorem 1.2.While it is believed that O ˚ p n q is the right bound for computing the chromatic number, we have no strongevidence to support this. There are reductions from popular problems and conjectures (like SETH) to otherpartitioning problems [6] or other parameterizations of the coloring problem [14]. It is interesting whether itcan be showed that an O ˚ pp ´ ε q n q algorithm for computing the chromatic number would refute any otherpopular conjecture. This question was raised several times, including in the book of Fomin and Kaski [10].Another technical contribution of the paper is Theorem 1.8. We believe that the presented removal lemmacould serve as a tool in the design of other exponential algorithms. It would be interesting to find moreproblems for which it can be used. 24 cknowledgements The author would like to deeply thank Noga Alon for important discussions and insights regarding the subsetremoval lemma, and Haim Kaplan and Uri Zwick for many helpful discussions and comments on the paper.
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A Appendix
A.1 Upper Bound for Section 5
In this section we provide a construction showing that in Theorem 1.8 we must have ρ p ∆ , C q ď p C ` q ´ ∆ .This bound is due to discussions with Noga Alon. Theorem A.1.
For any positive integers C, ∆ , n we can construct a collection F of p C ` q ∆ ¨ n sets ofsize ∆ , such that for every subsets F Ď F and U Ď U satisfying that @ F , F P F . F X F Ď U , we have | F | ´ C | U | ě n . T the complete p C ` q -ary tree of depth ∆ ´
1. Let the universe U be the set of T ’s vertices. Let F be the collection of p C ` q p C ` q ∆ ´ sets corresponding to root-to-leaf paths in T taken with multiplicity p C ` q each. Each set of F contains the ∆ vertices in its corresponding path, each such path has p C ` q identical sets corresponding to it in F . Lemma A.2.
For every F Ď F and U Ď U satisfying that @ F , F P F . F X F Ď U , we have | F | ´ C ¨ | U | ď .Proof. Denote by r the root of T . If r R U then | F | ď r . Otherwise, denote by T theconnected component of T r U s (i.e., the induced sub-graph of T on the vertex set U ) containing r . Denoteby (cid:96) the number of leaves in T , and by | T | the total number of vertices in T . As T is a p C ` q -ary tree,we have p C ` q ¨ (cid:96) ď ` C ¨ | T | . Consider a leaf v of T which is not a leaf of T . It has p C ` q childrenin T and by definition, all are not in U . Thus, at most one set in F can contain each of these children. Inparticular, at most p C ` q sets in F contain v . Consider a leaf v of T which is also a leaf of T . There is onlyone root-to-leaf path containing v , and it appears in F with multiplicity p C ` q . Hence, there are at most p C ` q sets in F containing v . From both cases we conclude that | F | ď p C ` q¨ (cid:96) ď ` C ¨| T | ď ` C ¨| U | .Thus, | F | ´ C ¨ | U | ď n disjoint copies of F over different base sets. A.2 Derandomizing Lemma 6.1
The construction of a dominating set in Lemma 6.1 can be done in a deterministic manner using the methodof conditional expectations ([29] [25]) as follows. We first note that, in the notation of Lemma 6.1, and forevery disjoint subsets V , V Ď V p G q , we can efficiently compute E r| R | | @ v P V .v P R ^ @ v P V .v R R s by using the linearity of expectation and considering the following cases: • If v P V then P r p v P R q “ • If v P V : – If N p v q X V ‰ H then P r p v P R q “ – Else,
P r p v P R q “ p ´ p q | N p v qz V | . • If v R V Y V : – If N p v q X V ‰ H then P r p v P R q “ p . – Else,
P r p v P R q “ p ` p ´ p q `| N p v qz V | .We next notice that if u R V Y V then E r| R | | @ v P V .v P R ^ @ v P V .v R R s “ p ¨ E r| R | | @ v P V Y t u u .v P R ^ @ v P V .v R R s`p ´ p q¨ E r| R | | @ v P V .v P R ^ @ v P V Y t u u .v R R s . Thus, we can add u into either V or V without increasing the above expectation. We therefore can iterativelyadd every vertex of V p G q to either V or V without increasing the conditional expectation. We finish witha concrete choice of R such that | R | is bounded by the original E r| R |s|s